Directions in Type I spaces

Abstract

A direction in a Type I space $X=\cup_{\alpha<\omega_1}X_\alpha$ is a closed and unbounded subset $D$ of $X$ such that given any continuous $f:X\to\mathbb{L}_{\ge 0}$ (the closed long ray), if $f$ is unbounded on $D$ then $f$ is unbounded on each unbounded subset of $D$. A closed copy of $\omega_1$ is a direction in any Type I space. We study various aspects of directions and show some independence results. A sample: There is an $\omega$-bounded Type I space without direction; PFA implies that a locally compact countably tight $\omega_1$-compact Type I space contains a direction; if there is a Suslin tree then there is an $\omega_1$-compact Type I manifold without direction; there are Type I first countable spaces which contain directions and whose closed unbounded subsets contain each a closed unbounded discrete subset. We also study a naturel order on the directions of a given space and show that we may obtain various classical ordered types with the space a manifold (often $\omega$-bounded).